Paper Review

Aims

1. Use gaussian processes (GPs) to resample Type II(b) SNe light curves
2. Estimate some morphological parameters as defined by Pessi et al. (2019)
   • Expecting a sharp divide between Type II and Type IIb SNe.
3. Assess the appropriateness of using GPs for this purpose
   • Goodness-of-fit of curves
   • Clustering of SNe by morphology parameters

Type IIb SNe Light Curves

Type II SNe Light Curves

Gaussian Process

• mean function, \(\mu(t)\)
• covariance or kernel function, \(k(t_i, t_j)\)

\[\boldsymbol{Y} = \begin{bmatrix} Y_1 \\ \vdots \\ Y_n \end{bmatrix} \sim \mathcal{GP}(\boldsymbol{\mu}, \boldsymbol{\Sigma})\] where \(\boldsymbol{\mu} = \mu(t_i),\; \boldsymbol{\Sigma} = \mathrm{Cov}(Y_i, Y_j) = k(t_i, t_j)\quad i,j = 1, \dots, n\).

\[\textrm{Squared Exponential} \qquad k(\tau; \lambda) = A \exp\left\{-\frac{1}{2}\left( \frac{\tau}{\lambda}\right)^2\right\}\]

\[\textrm{Matern-3/2}\qquad k(\tau; \lambda) = \left(1 + \sqrt{3}\left(\frac{\tau}{\lambda}\right)\right) \exp\left\{-\sqrt{3}\left(\frac{\tau}{\lambda}\right) \right\}\]

Dataset

Twenty-one “high quality” lightcurves from the Open Supernova Catalog accessible by API:

1. Evenly and densely sampled
2. Well-studied explosion
3. Chosen by visual inspection (!)

Methodology

1. Fit a GP using different kernels (RBF, Matern-3/2)

2. Visually assess the goodness-of-fit

   • mean function (peak, plateau, linear decay)

3. Estimate the morphology parameters

   • \(t_\textrm{rise}\): time between explosion to maximum light
   • \(\Delta m_{40-30}\): mag. difference between phase 30 and 40
   • dm1: the earliest maximum of first derivative
   • dm2: the earliest minimum of second derivative

Results

Matern 3/2 vs RBF

Different GP implementations

First and Second Derivatives

NB: Matern-3/2 processes are only 1-time differentiable.

Clustering by \(t_\textrm{rise}\) and \(\Delta m_{40-30}\)

Conclusions

• Kernel choice is crucial, especially length-scale.
• Adding kernels together can partially fit complex behaviours at different scales.
• SN light curves are perhaps better fitted using non-stationary kernels that allow varying smoothness.
• Be cautious of different software implementations of kernel turning resulting in different results.
• Results are heavily dependent on the density of sampling.
• Still reproduced the clustering by Pessi et al. (2019)

Statistical claims needing caveats

• GPs tend to overfit
  • easy to say when the physics and behaviour are known.
  • selection of curves and goodness-of-fit was judged by visual inspection without consideration of variances.
• Don’t use models outside the range of their training data
  • Depends on context, e.g., what about forecasting?
• GP interpolations not suited to estimating dm1 and dm2
  • Matern-\(\nu\) kernels are only differentiable (“smooth”) up to \(\nu-1\) derivative.

Things to try

• Non-stationary GP kernels
• Uneven or sparsely sampled SNe light curves
• Incorporate uncertainty of observations